# Properties

 Label 169.8 Level 169 Weight 8 Dimension 8081 Nonzero newspaces 8 Sturm bound 18928 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$169 = 13^{2}$$ Weight: $$k$$ = $$8$$ Nonzero newspaces: $$8$$ Sturm bound: $$18928$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_1(169))$$.

Total New Old
Modular forms 8395 8286 109
Cusp forms 8167 8081 86
Eisenstein series 228 205 23

## Trace form

 $$8081 q - 66 q^{2} - 66 q^{3} - 66 q^{4} - 66 q^{5} - 66 q^{6} + 2962 q^{7} - 9282 q^{8} + 11598 q^{9} + O(q^{10})$$ $$8081 q - 66 q^{2} - 66 q^{3} - 66 q^{4} - 66 q^{5} - 66 q^{6} + 2962 q^{7} - 9282 q^{8} + 11598 q^{9} + 19854 q^{10} - 7974 q^{11} - 69174 q^{12} - 15036 q^{13} + 21570 q^{14} + 82230 q^{15} + 163774 q^{16} + 131460 q^{17} - 499638 q^{18} - 307130 q^{19} + 212526 q^{20} + 684306 q^{21} + 438354 q^{22} - 29994 q^{23} - 1005282 q^{24} - 468804 q^{25} - 409542 q^{26} - 308502 q^{27} + 1363018 q^{28} - 153984 q^{29} + 1252446 q^{30} + 516034 q^{31} + 2101818 q^{32} - 73134 q^{33} - 1400178 q^{34} - 799326 q^{35} - 641994 q^{36} + 498412 q^{37} - 3950838 q^{38} + 368952 q^{39} - 5592654 q^{40} - 4346652 q^{41} - 1238214 q^{42} + 2983370 q^{43} + 13480362 q^{44} + 14018664 q^{45} + 12725670 q^{46} + 1041282 q^{47} - 9502434 q^{48} - 10941790 q^{49} - 15286974 q^{50} - 10405734 q^{51} - 6489706 q^{52} - 11394318 q^{53} - 10047966 q^{54} + 6655026 q^{55} + 25179498 q^{56} + 29264610 q^{57} + 31590894 q^{58} + 14215842 q^{59} + 20739378 q^{60} - 5194908 q^{61} - 427650 q^{62} - 49890318 q^{63} - 84290598 q^{64} - 25073793 q^{65} - 26946342 q^{66} + 20649838 q^{67} + 40618602 q^{68} + 40702986 q^{69} + 77131722 q^{70} + 39828438 q^{71} + 92094426 q^{72} + 25356742 q^{73} - 736218 q^{74} - 50339982 q^{75} - 118957802 q^{76} - 88013838 q^{77} - 85447650 q^{78} - 31689126 q^{79} - 18659646 q^{80} + 55314414 q^{81} + 105835158 q^{82} + 71699562 q^{83} + 161105370 q^{84} + 132164940 q^{85} + 122549154 q^{86} - 24279666 q^{87} - 100152342 q^{88} - 107770338 q^{89} - 256347246 q^{90} - 43758286 q^{91} - 176845950 q^{92} - 1203942 q^{93} + 30585270 q^{94} + 161418966 q^{95} + 400664730 q^{96} + 149722894 q^{97} + 95062182 q^{98} + 5830650 q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_1(169))$$

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space $$S_k^{\mathrm{new}}(N, \chi)$$ we list available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
169.8.a $$\chi_{169}(1, \cdot)$$ 169.8.a.a 1 1
169.8.a.b 2
169.8.a.c 4
169.8.a.d 6
169.8.a.e 8
169.8.a.f 8
169.8.a.g 14
169.8.a.h 21
169.8.a.i 21
169.8.b $$\chi_{169}(168, \cdot)$$ 169.8.b.a 2 1
169.8.b.b 4
169.8.b.c 8
169.8.b.d 14
169.8.b.e 16
169.8.b.f 42
169.8.c $$\chi_{169}(22, \cdot)$$ n/a 168 2
169.8.e $$\chi_{169}(23, \cdot)$$ n/a 170 2
169.8.g $$\chi_{169}(14, \cdot)$$ n/a 1260 12
169.8.h $$\chi_{169}(12, \cdot)$$ n/a 1248 12
169.8.i $$\chi_{169}(3, \cdot)$$ n/a 2544 24
169.8.k $$\chi_{169}(4, \cdot)$$ n/a 2520 24

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_1(169))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_1(169)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 2}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_1(169))$$$$^{\oplus 1}$$